I know that simple random walk (SRW) on a connected graph is irreducible. I understand why based on intuition and logic, but I'm having a hard time convincing myself mathematically. Is there a rigorous proof for this?
2026-04-03 20:58:21.1775249901
Simple Random Walk & Irreducibility
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start with the definition of two nodes communicating --i.e. non-zero probability of transitioning from some node $i$ to some node $j$ in finitely many states. For $i\lt j$, $k:= j-i$ and by inspection the probability of going from $i\to j$ in k steps is $2^{-k}$. By symmetry that is also the probability of going from $j\to i$ in exactly $k$ steps. A lot of this comes down to looking up and directly applying definitions.